When a deterministic approximation of \(_t\) , say \({\hat{}}_t\) , is employed in (REF ), which may come from a finite difference approximation of the second-order derivative, or from a quasi-Newton update such as BFGS or DFP, the convergence behavior is well understood. Specifically, if \(\lbrace {\hat{}}_t\rbrace _t\)  are positive definite with uniform upper and lower bounds, the damped phase is preserved by the same analysis as Newton's method. Furthermore, the quadratically convergent phase is weakened to a superlinearly convergent phase. This occurs if and only if the celebrated Dennis-Moré condition
[1]} holds, i.e.,
\(\lim \limits _{t\rightarrow \infty } \frac{\Vert ({\hat{}}_t - _t){\hat{}}_t^{-1}\nabla f_t\Vert }{\Vert {\hat{}}_t^{-1}\nabla f_t\Vert } = 0.\) 
